This invention relates to time series analysis. Time series analysis and forecasting or extrapolating of future values is one of the basic tasks in quantitative analytics. Important application areas are census and econometrics: predicting national income, unemployment rates, birth rates, and the like; logistics and planning: predicting sales volumes, income, cost, price; medicine: predicting medical state of patients; or science: weather forecast, climate, global warming, and the like.
In the state of the art, a wide range of mathematical methods and algorithms have been proposed, such as seasonal-trend models, seasonal autoregressive integrated moving average (ARIMA) models, or the so-called exponential smoothing models. The basic idea of exponential smoothing (without seasonality and trend effects in the time series date to be considered) is to predict future values ypred(t+1) at time t+1 of a time series y(t) at time t, based on the series' known values from the past, with decreasing importance of less recent values:ypred(t+1):=αy(t)+(1−α)ypred(t)=αΣk=0 . . . N−1(1−α)ky(t−k) with damping factor 0<α<1.
With trend and additive season comprise in the time series data, the so-called Holt-Winter model is obtained:yt=gt+st+et     with gt=α·(yt−st−s)+(1−α)·(gt−1+φ·Tt−1) being the general trend; (φ˜0.9 . . . 0.99 is trend damping);    with Tt=β·(gt−gt−1)+(1−β)·φ·Tt−1 being the incremental trend component at time t    and st=δ·(yt−gt−1)+(1−δ)·st−s being the seasonal component at time t.For parameter initialization: e.g. α,β,δ=0.1 . . . 0.3; φ=0.9 . . . 0.99; g0=y1; T0, sk<1=0 is used.
Additive trend/additive season (AA), no trend/no season (NN), no trend/additive season (NA), additive trend/no season (AN) are also covered by this model.
More generally, Pegels and Gardner have proposed a classification scheme which groups different exponential smoothing models by:{kind of trend}*{additive or multiplicative season}
As a recursive procedure, an exponential smoothing data analysis process requires some computational effort to try to fit all the model types. Such a procedure could cause problems in terms of overfitting the data rather than finding the true nature of the data generating process. An experienced user could be able to select a model type by means of graphical analysis of a time series plot, but an inexperienced user might have some problems to make the right decisions for using formalized, stable and reliable criteria.
An overview of existing automated model selection methods has been given by E. S. Gardner Jr. published in Exponential Smoothing: The state of the art—Part II, 2005, http://www.bauer.uh.edu/gardner/. Gardner and McKenzie in Gardner E. S. Jr., McKenzie, Model identification in exponential smoothing, Journal of the Operational Research Society, 1988, no. 39, pp. 863-867, suggest a method of minimizing the variance of transformed time series. However, only additive trends and multiplicative seasonality are considered there. This approach was tested empirically by Tashman and Kruk, published in Tashman L. J., Kruk J. M., The use of protocols to select exponential smoothing procedures: A reconsideration of forecasting competitions, International Journal of Forecasting, 1996, no. 12, pp. 235-253 and Taylor in Taylor J. W., Exponential smoothing with a damped multiplicative trend, International Journal of Forecasting, 2003, no. 19, pp. 715-725. Alternative methods were developed by Shah, published in Shah C., Model selection in univariate time series forecasting using discriminant analysis, International Journal of Forecasting, 1997, no. 13, pp. 489-500: only two model types, without trend and seasonality as well as additive trend and multiplicative seasonality, are considered; and Meade published in Meade N., Evidence for the selection of forecasting methods, Journal of Forecasting, 2000, no. 19, pp. 515-535: no trend and seasonality, additive trend and no seasonality, damped additive trend and no seasonality have been considered therein. In these cases, there were also non-smoothing competing methods taken into account.
Optimizing the parameters is suggested by Hyndman as published in Hyndman R. J., Wheelwright S. C., Makridakis S., Forecasting: methods and applications, 3rd ed., 1998, John Wiley & Sons, New York. However, it is not defined which optimizing procedure is best for this purpose.